Skip to main content

Orthogonality — Conventional and Unconventional — In Numerical Analysis

  • Chapter
Computation and Control

Part of the book series: Progress in Systems and Control Theory ((PSCT,volume 1))

Abstract

The idea of orthogonality is widespread in numerical analysis. In its analytic form, it is used to great advantage in problems of least squares approximation, quadrature, and differential equations. The principal tools are orthogonal polynomials. In its algebraic (finite-dimensional) form, orthogonality underlies many iterative methods for solving large systems of linear algebraic equations. If used in similarity transformations, it leads to effective methods of computing eigenvalues. Here, we shall limit ourselves to two application areas: numerical quadrature and univariate approximation. We review a number of applications in which orthogonality plays a significant role and in some of which nonstandard features suggest interesting new problems of analysis and computation.

Work supported, in part, by the National Science Foundation under grant CCR-8704404.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. B. Baillaud and H. Bourget, Correspondance d’Hermite et de Stieltjes I, IL Gauthier-Villars, Paris, 1905.

    Google Scholar 

  2. S. Bernstein, “Sur les polynomes orthogonaux relatifs a un segment fini”, J. Math. Pures Appl. (9), v. 9,1930, pp. 127–177.

    Google Scholar 

  3. F. Calio, W. Gautschi and E. Marchetti, “On computing Gauss-Kronrod quadrature formulae”, Math. Comp., v. 47, 1986, pp. 639–650.

    Google Scholar 

  4. L. Chakalov , “General quadrature formulae of Gaussian type” (Bulgarian), Bulgar. Akad. Nauk Izv. Mat. Inst., v.1, 1954, pp. 67–84.

    Google Scholar 

  5. L. Chakalov “Formules générales de quadrature mécanique du type de.Gauss”, Colloq. Math., v. 5, 1957, pp. 69–73.

    Google Scholar 

  6. P.L. Chebyshev, “Sur rinterpolation par la méthode des moindres carrés”, Mem. Acad. Impér. Sei. St. Pétersbourg (7), v.1, no.15, 1859, pp. 1–24. [Oeuvres I, pp. 473–498.].

    Google Scholar 

  7. T.S. CHIHARA, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.

    Google Scholar 

  8. E.B. Christoffel, “Über die Gaußische Quadratur und eine Verallgemeinerung derselben”, J. Reine Angew. Math., v. 55, 1858, pp. 61–82. [Ges. Math. Abhandlungen I, pp. 65–87.].

    Article  MATH  Google Scholar 

  9. O. Egeciogluand C.K. Koc, “A fast algorithm for rational interpolation via orthogonal polynomials”, Math. Comp., v. 53,1989, to appear.

    Google Scholar 

  10. S. Elhay and J. Kautsky, “A method for computing quadratures of the Kronrod Patterson type”, Austral. Comput. Sei. Comm., v. 6, 1984, no.1, pp. 15.1–15.9. Department of Computer Science, University of Adelaide, Adelaide, South Australia.

    Google Scholar 

  11. G.E. Forsythe, “Generation and use of orthogonal polynomials for data-fitting with a digital computer”, J. Soc. Indust. Appl. Math., v. 5, 1957, pp. 74–88.

    Article  MathSciNet  MATH  Google Scholar 

  12. G. Freud, Orthogonal Polynomials, Pergamon Press, New York, 1971.

    Google Scholar 

  13. L. FREY, Extension of Quadrature Formulas - Theory and Numerical Methods, Ph.D. Thesis, Federal Institute of Technology (ETH), Zürich, in preparation.

    Google Scholar 

  14. M. Frontini, W. Gautschi and G.V. Milovanovic, “Moment-preserving spline approximation on finite intervals”, Numer. Math., v. 50, 1987, pp. 503–518.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Galant, “An implementation of Christoffers theorem in the theory of orthogonal polynomials”, Math. Comp., v. 25,1971, pp. 111–113.

    MathSciNet  MATH  Google Scholar 

  16. W. Gautschi, “On generating Gaussian quadrature rules”, in: Numerische Integration (G. Hammerlin, ed.), pp. 147–154, Internat. Ser. Numer. Math., v. 45. Birkhäuser, Basel, 1979.

    Google Scholar 

  17. W. Gautschi, “A survey of Gauss-Christoffel quadrature formulae, in:E.B. Christoffel” The Influence of his Work in Mathematics and the Physical Sciences (P.L. Butzer & F. Féher, eds.), pp. 72–147. Birkhäuser, Basel, 1981.

    Google Scholar 

  18. W. Gautschi, “An algorithmic implementation of the generalized.Christoffel theorem”, in: Numerical Integration (G. Hämmerlin, ed.), pp. 89–106, Internat. Ser. Numer. Math., v. 57. Birkhäuser, Basel, 1982

    Google Scholar 

  19. W. Gautschi, “On generating orthogonal polynomials”, SIAM J. Sei. Sta- tist. Comput., v. 3,1982, pp. 289–317.

    Article  MathSciNet  MATH  Google Scholar 

  20. W. Gautschi, “Discrete approximations to spherically symmetric distributions”, Mwier. Math., v. 44,1984, pp. 53–60.

    Google Scholar 

  21. W. Gautschi, “Questions of numerical condition related to polynomials”in: Studies in Numerical Analysis (G.H. Golub, ed.), pp. 140–177, Studies in Mathematics, v. 24. The Mathematical Association of America, 1984.

    Google Scholar 

  22. W. Gautschi, “On the sensitivity of orthogonal polynomials to perturbations in the moments”, Numer. Math., v. 48,1986, pp. 369–382.

    Google Scholar 

  23. W. Gautschi, “Gauss-Kronrod quadrature - A survey”, in: Numerical.Methods and Approximation Theory III (G.V. Milovanovic, ed.), pp. 39–66. Faculty of Electronic Engineering, University of Nis, Nis, 1988.

    Google Scholar 

  24. W. Gautschi , On the zeros of polynomials orthogonal on the semicircle.SIAM J. Math. Anal, v. 20,1989, to appear.

    Google Scholar 

  25. W. Gautschi and G.V. Milovanovic, “Spline approximations to spheri- cally symmetric distributions”, Numer. Math., v. 49,1986, pp. 111–121.

    Google Scholar 

  26. W. Gautschi and G.V. Milovanovic, “Polynomials orthogonal on the semicircle”.>J. Approx. Theory, v. 46, 1986, pp. 230–250.

    Google Scholar 

  27. G.V. Milovanovic and S.E. Notaris, “An algebraic study of Gauss-Kronrod.quadrature formulae for Jacobi weight functions”, Math. Comp., v. 51, 1988, pp. 231–248.

    Google Scholar 

  28. G.V. Milovanovic and S.E. Notaris, “Gauss-Kronrod quadrature formulae for.weight functions of Bernstein-Szegö type”, J. Compiu. Appl Math., v. 25,1989, to appear.

    Google Scholar 

  29. S.E. Notaris and TJ. Rivlin, “A family of Gauss-Kronrod quadrature.formulae”, Math. Comp., v. 51, 1988, pp. 749–754.

    Google Scholar 

  30. HJ. Landau and G.V. Milovanovic, “Polynomialsorthogonal on the semicircle II”, Constructive Approx., v. 3, 1987, pp. 389–404.

    Google Scholar 

  31. G.H. Golub and J.H. Welsch, “Calculation of Gauss quadrature rules”, Math. Comp., v. 23, 1969, pp. 221–230. Loose microfiche suppl. A1–A10.

    Google Scholar 

  32. P.R. Graves-Morris and T.R. Hopkins, “Reliable rational interpolation”, Numer. Math., v. 36, 1981, pp. 111–128.

    Google Scholar 

  33. A.S. KRONROD, Nodes and Weights for Quadrature Formulae. Sixteen-place Tables (Russian). Izdat. “Nauka”, Moscow, 1964. [English translation: Consultants Bureau, New York, 1965.].

    Google Scholar 

  34. J.-C. LIN, Rational L2-Approximation with Interpolation, Ph.D. Thesis, Purdue University, 1988.

    Google Scholar 

  35. CA. Michelli, “Monosplines and moment preserving spline approximation”, in: Numerical Integration III (H. Braß and G. Hämmerlin, eds.), pp. 130–139, Internat. Ser. Numer. Math., v. 85. Birkhäuser, Basel, 1988.

    Google Scholar 

  36. G.V. Milovanovic , “Construction of orthogonal polynomials and Turin quadrature formulae, in: Numerical Methods and Approximation Theory III” (G.V. Milovanovic, ed.), pp. 311–328. Faculty of Electronic Engineering, University of Nis, Nis, 1988.

    Google Scholar 

  37. and M.A. KOVACEVIC, “Moment-preserving spline approxi- mation and Turan quadratures”, in: Numerical Mathematics: Singapore 1988 (R. Agarwal, Y. Chou and S. Wilson, eds.), pp. 357–365, Internat. Ser. Numer. Math., v. 86. Birkhäuser, Basel, 1988.

    Google Scholar 

  38. G. Monegato, “A note on extended Gaussian quadrature rules”, Math. Comp., v. 30,1976, pp. 812–817.

    Google Scholar 

  39. G. Monegato, “Positivity of the weights of extended Gauss-Legendre qua- drature rules”, Math. Comp., v. 32,1978, pp. 243–245.

    Google Scholar 

  40. G. Monegato, “On polynomials orthogonal with respect to particular.variable-signed weight functions”, Z. Angew. Math. Phys., v. 31, 1980, pp. 549–555.

    Google Scholar 

  41. G. Monegato , “Stieltjes polynomials and related quadrature rules”, SI AM. Rev., v. 24,1982, pp. 137–158.

    Google Scholar 

  42. LP. Mysovskih, “A special case of quadrature formulae containing preassigned nodes” (Russian), Vesci Akad. Navuk BSSR Ser. Fiz.-Tehn. Navuk, no.4,1964, pp. 125–127.

    Google Scholar 

  43. S.E. Notaris, “Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegö type II”, /. Comput. Appl. Math., to appear.

    Google Scholar 

  44. A. Ossicini, “Costruzione di formule di quadratura di tipo Gaussi-ano”, Ann. Mat. Pura Appl. (4), v. 72, 1966, pp. 213–237.

    Google Scholar 

  45. T. Popoviviu, “Sur une generalisation de la formule d’intégration numérique de Gauss”, Acad. R.P. Ronane Fil. Ia$i Stud. Cerc.gti., v. 6, 1955, pp. 29–57.

    Google Scholar 

  46. R. , “Etude sur les formules d’approximation qui servent a cal-culer la valeur numérique d’une integrale définie” J. Math. Pures Appl. (3), v. 6, 1880, pp. 283–336

    Google Scholar 

  47. D.D. Stancu, Generalization of the quadrature formula of Gauss-Christoffel (Romanian), Acad. R.P. Romxne Fil. Ia§i Stud. Cere. §ti. Mat., v. 8, no.l, 1957, pp. 1–18.

    Google Scholar 

  48. D.D. Stancu , “On a class of orthogonal polynomials and on some general.quadrature formulae with minimum number of terms” (Romanian), Bull. Math. Soc. Sei. Math. Phys. R.P. Roumaine (N.S.), v.1, no.49, 1957, pp. 479–498.

    MathSciNet  Google Scholar 

  49. D.D.Stancu , “Sur quelques formules générales de quadrature du type.Gauss-Christofifel”, Mathematica (Clujh v.1 (24), 1959, pp. 167–182.

    Google Scholar 

  50. E.L. Stiefel, “Kernel polynomials in linear algebra and their numerical applications” in: Further Contributions to the Solution of Simultaneous Linear Equations and the Determination of Eigenvalues, NBS Applied Math. Ser., v. 49, 1958, pp. 1–22.

    MathSciNet  Google Scholar 

  51. T.J. Stieltjes, “Quelques recherches sur la théorie des quadratures dites mécaniques”., Ann. Sei. Ecole Norm. Paris, Sér. 3, v.1, 1884, pp. 409–426. [Oeuvres I, pp. 377–396.].

    Google Scholar 

  52. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer, New York, 1980.

    Google Scholar 

  53. G. SZEGÖ, “Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören”, Math. Ann., v. 110, 1935, pp. 501–513. [Collected Papers II, pp. 545–557.].

    Google Scholar 

  54. P. Turan, “On the theory of the mechanical quadrature”, Acta Sei. Math. Szeged, v. 12,1950, pp. 30–37.

    Google Scholar 

  55. J. WALDVOGEL and L. FREY, “Repeated extension of Gaussian quadrature formulas by means of eigenvalue techniques”, in preparation.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Birkhäuser Boston

About this chapter

Cite this chapter

Gautschi, W. (1989). Orthogonality — Conventional and Unconventional — In Numerical Analysis. In: Computation and Control. Progress in Systems and Control Theory, vol 1. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3704-4_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-3704-4_5

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-0-8176-3438-4

  • Online ISBN: 978-1-4612-3704-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics